# \int_{k}^{\infty} e^{-a^{2}} dy (gauss int. w. finite lim.)

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• May 29th 2008, 02:30 PM
AA1983
\int_{k}^{\infty} e^{-a^{2}} dy (gauss int. w. finite lim.)
I am doing a calculation of overlap between wavefunctions in a physics thesis, however this integral puzzles me:

$\int_{-L/2+\kappa a^{2}}^{\infty} e^{-\frac{1}{2a^{2}}y^{2}} dy$

From a physic point of view it is very important that the integral is from $-L/2+\kappa a^{2}$ and not from $-\infty$. I am awere that is is impossible to write down an antiderivative of the function being integrated however, there must be some way to deal with the integral.

How do I calculate this integral (as a function of momentum $\kappa$)?
• May 29th 2008, 02:54 PM
wingless
$D=-L/2+\kappa a^{2}$

$\int_{D}^{\infty} e^{-\frac{y^2}{2a^{2}}} dy$

$\int_{D}^{\infty} e^{-\left ( \frac{y}{a\sqrt{2}}\right )^2} dy$

Let $x =\frac{y}{a\sqrt{2}}$. Then $dy = a\sqrt{2}~dx$.

The bounds are now from $\frac{D}{a\sqrt{2}}$ to infinity.

It becomes $\int_{\frac{-L/2+\kappa a^{2}}{a\sqrt{2}}}^{\infty}e^{-x^2}~dx$.

$\int_{\frac{-L/2+\kappa a^{2}}{a\sqrt{2}}}^{\infty}e^{-x^2}~dx = \int_{0}^{\infty}e^{-x^2}~dx - \int_{0}^{\frac{-L/2+\kappa a^{2}}{a\sqrt{2}}}e^{-x^2}~dx$

$\int_{0}^{\infty}e^{-x^2}~dx$ is the Gaussian integral and $\frac{\sqrt{\pi}}{2}$

$\frac{\sqrt{\pi}}{2} - \int_{0}^{\frac{-L/2+\kappa a^{2}}{a\sqrt{2}}}e^{-x^2}~dx$

$\frac{\sqrt{\pi}}{2} - \frac{\sqrt{\pi}}{2} \mbox{Erf}(\frac{-L/2+\kappa a^{2}}{a\sqrt{2}})$

You have to use an approximation of the error function here.
An approximation of Erf is: $|\mbox{Erf}(x)| = \sqrt{1-e^{-(2x/\sqrt{\pi})^2}}$

Another one (and a better one, I think) is here: Error function - Wikipedia, the free encyclopedia
• May 29th 2008, 03:26 PM
AA1983
wingless, thanks a lot. I really appreciate it.

I've never come across error functions in this context if ever at all. Seems like I have only done easy -inf to inf gaussians in electro class.
• May 29th 2008, 03:31 PM
wingless
Quote:

Originally Posted by AA1983
wingless, thanks a lot. I really appreciate it.

I've never come across error functions in this context if ever at all. Seems like I have only done easy -inf to inf gaussians in electro class.

Not at all.. If you are looking for more about the error function, you can see the wikipedia thread (Sun)